Inverse Trigonometric Functions

Like sine function, the cosine function is a function whose domain is the set of all real numbers and range is the set [-1, 1]. If we restrict the domain of cosine function to \([0, \pi]\), then it becomes one-one and onto with range [-1, 1]. Actually, cosine function restricted to any of the intervals \([-\pi, 0]\), \([0, \pi]\), \([\pi, 2\pi]\) etc., is bijective with range as [-1, 1]. We can, therefore, define the inverse of cosine function in each of these intervals. We denote the inverse of the cosine function by \(cos^{-1}\) (arc cosine function). Thus, \(cos^{-1}\) is a function whose domain is [-1, 1] and range could be any of the intervals \([-\pi, 0]\), \([0, \pi]\), \([\pi, 2\pi]\) etc. Corresponding to each such interval, we get a branch of the function \(cos^{-1}\). The branch with range \([0, \pi]\) is called the principal value branch of the function \(cos^{-1}\). We write \(cos^{-1}: [-1, 1] \rightarrow [0, \pi]\).